Weak disord.:

Strong disord.:

Outline

• Experiments on graphene nanoribbons GNR (IBM and Columbia, 2007) • Model of weakly disordered ribbons (solvable but artificial, somewhat) • Model for strongly disordered ribbons (interesting but hard) • Ballistic transport through metal-graphene-metal junctions – “contact-independent” conductance

Graphene in the grand scheme of things

Borrowed from Geim’s 2007 KITP presentation

Fabrication and field effect

K’ K

Geim and Novoselov, Nature Materials, 2007

: BZ

Ribbons

20 nm 30 nm 40 nm 50 nm 100 nm 200 nm

Chen, Lin, Rooks, and Avouris, cond-mat 2007

Han, Özyilmaz, Zhang, and Kim, cond-mat 2007

I(T) and current 1/f noise

e

− E g / 2T

?

Chen, Lin, Rooks, Avouris (IBM), cond-mat 2007

Width-dependence of gap, Eg(W)

Eg =

α W − W0

Han, Özyilmaz, Zhang, and Kim, cond-mat 2007

Questions to be answered

• • • • •

What causes activated behavior? Why at T= 4K conductivity is relatively high? [Is it a real semiconducting gap?] How can gap depend on the width?! What causes 1/f noise: background charge fluctuations or intrinsic fluctuations?

What is known: clean perfect ribbons

• zig-zag are gapless (edge modes) • armchairs: gapped or gapless W = 3N, 3N+2

W = 3N+1

Ezawa, PRB 2006; Brey-Fertig, PRB 2006

Zig-zag ribbons and zig-zag edges Zero-energy evanescent states in semi-infinite graphene: ⎛ 0 ⎜ ⎝ px − ip y

Edge:

⎛ e − k y x +ik y y ⎞ ⎜⎜ ⎟⎟ ⎝ 0 ⎠

px + ip y ⎞ ⎛ a ⎞ =0 ⎟ 0 ⎠ ⎜⎝ b ⎟⎠ and

⎛ 0 ⎞ ⎜⎜ k y x +ik y y ⎟⎟ ⎝e ⎠

• Large ky – strong localization • In ribbon edge states interact x

NAKADA, FUJITA, DRESSELHAUS, AND DRESSELHAUS, PRB 1996

unhappy

happy

Armchair ribbons

Ribbon crosssection

Simplest model of disorder variable-width armchair GNR

~W

~L

~L

~L

~L

Take L >> W Metal – level spacing t/L Insulator – gap t/W

Band structure and Wave-function segmentation!

Example (Eg = 2 x 0.17 t)

(Eg = 2 x 0.17 t)

Eg

E = -0.1 t

E = -0.2 t

E = -0.4 t

Effective model 1D hopping tij

εi-1

εi

εj

εi

εi+1

εi+2

WKB: tij ~ e− 2m*U r

U ~ t /W ,

massin insulator :

ε ins =

Δεi ~ t/Li

U2+

p2vF2

2 p =U + 2(U / vF2 )

thus m* ~ (tag2W )−1 and

−rij /W

tij ~ e

2U

Hopping transport

σ ∼e

− rij / W −|ε i −ε j |/ T

for a hop of length

n: rij = nLav and | ε i − ε j |~ (t / Lav ) / n

Minimize over n n > 1: VRH av n = 1: NNH Crossover at

Hopping transport regimes

av log σ

~1/T ~1/ T Lav t

L2av tW

1/T

Strong disorder case

E = -0.07 t

E = -0.09 t

E = -0.255 t

Still segmented but with different degree of localization!

Strong disorder case – schematic

States with E < t/W

Counting states:

per plaquette W × W dos 2D × (t / W ) × W 2 ~ 1 state / plaquette Which states contribute to conductivity?

only “good” fat states All states

Two possibilities

W 1

W

W 2

W2

Compare with expt

e

− E g / 2T

?

log σ NNH ~ 1/T

~ 1/ T

Chen, Lin, Rooks, Avouris (IBM), cond-mat 2007

W t

W 2 t

VRH

1/T

Not all low energy states matter!

How about Coulomb? Efros-Shklovskii: extra Coulomb cost, e2/r

r – length of hop

e 2 / tag ~ 1 Graphene fine-structure constant Thus: • For long-hops via “good” states Coulomb is irrelevant • For short hops via all low energy states can be relevant

Can be tested by placing a metallic gate to screen Coulomb

Metal-Graphene-Metal (MGM)

Transport through NGN structure ts

t'

tg

tg

t'

ts

Indep of β!

t '2 β= t g ts V qg = t g ag

Conclusions

• Simple model for disordered nanoribbons • Low temperature transport in ribbons can be in variable or nearest neighbor hopping regimes • Coulomb appears irrelevant for the hopping through good fat states • 1/f noise appears to be consistent with hopping conductivity • NGN structure, resonances, and contact-independent resistance